Allie DeRienzo loves to dance, sing and play with her big brother, switching from the pink-and-purple cartoon world of Shimmer and Shine to the action-packed fantasy of Star Wars with the blink of an eye. It’s a flexibility that has served her well: In just a few years, she’s endured more ups and downs than most 3-year-olds.

Although her pregnancy was normal, it became clear as soon as Allie was born that something wasn’t right. “She was incredibly distended and was transferred almost immediately to a high-level neonatal intensive care unit (NICU) in New York,” remembers her mother, Nanci. …